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We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $Kmathrm{End}(mathcal{C})$ to topological Hochschild homology $mathrm{THH}(mathcal{C})$ for any spectral Waldhausen category $mathcal{C}$. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in $mathcal{C}$ indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausens $S_{bullet}$-construction for spectral Waldhausen categories, which is central to this account of the Dennis trace map. Our goals are both convenience and transparency---we provide all details except for a proof of the additivity theorem for $mathrm{THH}$, which is taken for granted---and the exposition is concerned not with originality of ideas, but rather aims to provide a useful resource for learning about the Dennis trace and its underlying machinery.
It is known by results of Dyckerhoff-Kapranov and of Galvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of
In a previous paper, we showed that a discrete version of the $S_bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, b
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